PART SECOND.

SECTION I.—Of Planes, and of Bodies terminated by Planes.

195. In the preceding sections, we have considered all the lines of any one of the figures which have been discussed, as in a plane surface; that is, in a surface to which a straight line may be applied in every direction, so as to lie entirely in this surface. This has enabled us to unfold the elementary principles of linear geometry, and to measure the areas of surfaces bounded by straight lines or circular curves. We proceed now to investigate the geometrical relations in bodies.

As the surfaces of many bodies consist, in whole or in part, of planes; and as all bodies involve the three dimensions of space, length, breadth and thickness; in discussing their geometrical character, it is necessary to consider not only the forms and magnitudes of planes, but their relative positions, their inclinations, and their relation to lines and points without them.*

In discussing the general relations of lines to planes, and of planes to each other, we consider them as indefinitely extended, except where some limit is expressly stated.

196. If two straight lines are in the same plane, a second plane may be conceived to pass through one of these lines without coinciding with the other line; it will therefore cut the first plane. But if any two points

* In what follows, the figures though represented on a plane, are considered as embracing the three dimension of space. And where any line is supposed to pass behind any part of a body or of a plane, it is indicated in the figure by a dotted line.

common to the two planes, be taken, a straight line drawn between them will lie wholly in each of the planes (8); therefore-The intersection of two planes is a straight line.

197. If the second plane be turned about the straight line through which it passes, till it coincides with the other line, it must also coincide with the other plane : Two straight lines, therefore, are sufficient to determine the position of a plane: And three points, not in the same straight line, will also determine the position of a plane; for the plane may be conceived to revolve about a straight line passing through two of the points; and if it be placed so as to coincide with the other point, it can revolve no farther without leaving it; it is therefore fixed.

It follows from this that any two straight lines which cut each other, are in the same plane; for a plane may be passed through one of them, as AB (fig. 108), and Fig.108. turned till it coincide with the other CD.

It is evident, therefore, that—any three points, not in the same straight line, being joined, two and two, by three straight lines, a plane triangle will be formed; if four points are so joined, the quadrilateral, thus formed, may be a plane figure; but if the plane passing through three of the points, do not at the same time pass through the fourth, the quadrilateral cannot be considered as a single surface.

19S. A straight line is said to be parallel to a plane, when it does not incline towards the plane, in either direction. A line parallel to a plane will therefore be, in every part, at the same distance from the plane; and consequently can never meet it, however far produced.

A straight line is perpendicular to a plane when its inclination to the plane is the same on all sides. It is evident that the straight line will, in this case, be perpendicular to every straight line drawn in this plane, through the foot of the perpendicular.

199. Let us suppose a straight line PD (fig. 109) Fig.109. to revolve about the straight line AB, always perpendicular to it, AB being considered as fixed. It is evident that PD, by this revolution, will generate a plane surface; for in whatever position the revolving line be placed, in the position PE, for instance, a plane may be conceived to pass through EPA, which being extended

Fig.109. will also pass through the opposite position of the revolving line, as PC. But as this line must be always perpendicular to AP, the sum of the two angles APE, APC, will be equal to two right-angles, and CPF will always be a straight line, that is, the surface will be a plane, to which AB will be perpendicular (8).

It follows, therefore, that—If two straight lines are perpendicular to the same straight line, at the same point, they are in the same plane perpendicular to this line.

200. Remark. It is evident that-There can be only one line passing through P, perpendicular to this plane; for if there could be another it must diverge from PA, and therefore its inclination to the plane must be greater on one side than the other, which cannot be the case with a perpendicular (198).

201. It is also evident that-Through any point, without a plane, as A for instance above the plane MN, only one perpendicular can be drawn to that plane. For a straight line passing through A in the same direction with AP must be the same line; another line, therefore, which passes through A must be inclined to AP, and consequently will be differently inclined to the plane. MN; it cannot, therefore, be perpendicular to this plane.

It is equally evident that-Only one plane can cut a straight line at the same point, perpendicular to it. For two planes passing through the same point, must be inclined to each other; they cannot, therefore, be both perpendicular to the same straight line.

202. As any two of the lines, CE, FD, passing through the point P in this plane, are sufficient to fix the position of this plane with respect to the line AB perpendicular to these two lines (197); we say—If a straight line is perpendicular to each of two straight lines drawn in a plane through its foot, it is perpendicular to every other straight line drawn through its foot in this plane, and is therefore perpendicular to the plane.

203. Suppose AP to be perpendicular to the plane Fig.110. MN (fig. 110); draw in this plane, from the foot of this perpendicular, the equal straight lines PC, PD; and draw AC, AD. The two triangles APC, APD, are both right-angled at P; the sides containing the equal angles are equal respectively; the triangles are therefore equal by the second case, and give ACAD. We therefore

say-Oblique_lines drawn to a plane from any point without it and equally distant from a perpendicular to the plane, drawn through the same point, are equal.

204. It follows from the last result, that-If from any point without a plane, a straight line be drawn perpendicular to that plane, and also several equal oblique lines be drawn to the plane from this point; these oblique lines will meet the plane in the circumference of a circle whose centre is the foot of the perpendicular. Consequently-If a straight line be drawn perpendicular to a circle through its centre, any point in this straight line is equally distant from every point in the circumference of the circle.

205. If we produce PD to G, and draw AG, it will be easy to show that AG must be greater than AD (51), and thence to show that-Of two oblique lines falling upon a plane at unequal distances from a perpendicular, that is the greater which falls at the greater distance from the perpendicular.

206. Suppose the line AG (fig. 111) to be oblique to the plane MN; draw from A to the plane the perpendicular AP; join PG, and through G draw BC perpendicular to PG. Then taking GC equal to GB, draw PB, PC, AB, AC; PB and PC will be equal (41), and AB and AC will therefore be equal; and AG will be a straight line drawn from the vertex of an isosceles triangle to the middle of the base; that is—BC is perpendicular to the oblique line AG, when it is perpendicular to the straight line which joins the foot of the oblique line with the foot of the perpendicular AP.

In this case AP and BC are said to be perpendicular to each other, though they cannot meet.

Fig.111.

207. Suppose the plane OP (fig. 112) to cut the plane Fig.112 MN, in the line AB; draw through the point C, CD in the plane MN, and CE in the plane OP, each perpendicular to the intersection AB of these planes; then turning the plane OP about the intersection till the two planes coincide, CE will coincide with CD. If the part E of the plane OP be now raised from the plane MN, turning this plane about the intersection, this point will describe the arc DE, which in every stage of the process will measure the inclination of the two planes; the centre of this arc is C; therefore-To measure the angle which two planes make with each other, draw in each plane, through the same point, a straight line perpendicular to the intersec

Fig.113.

tion, and the angle which these two lines make with each other is the angle of the planes.

208. Remark. The angle made by two planes is called a diedral angle, that is, an angle of two faces; and is designated by four letters, of which the two middle ones are in the intersection, and the two others are in the different planes out of the intersection; as the diedral angle PABN. The intersection AB of the two planes is called the edge of the diedral angle. When this angle is 90°, the inclination of the plane PO to the plane MN, is the same on each side of the intersection, and the planes are said to be perpendicular to each other.

It is evident that two diedral angles are to each other as the arcs of the plane angles which are their measures. It is also manifest that two diedral angles which are opposite at the edge, as MABO and PABN, are equal.

209. Suppose the two planes CD and EF (fig. 113) to be each perpendicular to the plane AB; from the point G where the three planes meet, draw EH perpendicular to the plane AB; it will be in each of the planes CD and EF (207); therefore-The intersection of two planes each perpendicular to a third plane, is also perpendicular to this third plane.

210. If two straight lines are perpendicular to the same plane, they have the same direction in space and are therefore parallel to each other. Let CE be perpenFig.114. dicular to the plane MN (fig. 114), and BD be parallel to CE; let the plane AD pass through the parallels; this plane will be perpendicular to the plane MN (207), and DB parallel to EC, will be perpendicular to the intersection AB; it will therefore have the same inclination to the plane MN, as the plane AD has to the plane MN; and as this inclination is a right angle, DB is perpendicular to the plane MN. Therefore-A straight line parallel to a second which is perpendicular to any plane, is also perpendicular to this plane.

Fig.115.

211. Take the two planes AB and CD (fig. 115) perpendicular to the same straight liue GH; draw GR and GL in the plane AB, and from the point H draw HM and HS parallel respectively to GL and GR; GL and GR will be both perpendicular to the line GH, and consequently will lie in the plane AB (201). Now the two lines HM and HS determine the position of the plane CD (198), and GL and GR determine the position of the